Smoothing Vertex-degree Bounded Polyhedra - Semantic Scholar
Purdue University
Purdue e-Pubs Computer Science Technical Reports
Department of Computer Science
1993
Smoothing Vertex-degree Bounded Polyhedra Jörg Peters Report Number: 93-051
Peters, Jörg, "Smoothing Vertex-degree Bounded Polyhedra" (1993). Computer Science Technical Reports. Paper 1065. http://docs.lib.purdue.edu/cstech/1065
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information.
SMOOTHING VERTEX·DEGREE BOUNDED POLYHEDRA
Jorg Peters
CSD-TR-93-6S1 August 1993
Smoothing vertex-degree bounded polyhedra
by Jorg Peters t [email protected]
Key words: C 1 surface, corner cutting, box splines, C 1 spline mesh, blending Running title: Smoothing polyhedra Version: Sept 15 93
Date printed: October 13, 1993
Submitted to: ACM TOG
Abstract A special class of meshes for outlining a surface, vertex-degree bounded, polyhedral meshes, is shown to have a simple smoothing algorithm that generates a small number of low degree polynomial pieces per cell. In particular, the input mesh need not be refined to obtain the Bernstein-Bezier control points as averages of the mesh points. Thus vertexdegree bounded, polyhedral meshes are the simplest and immediate generalization of the regular (tensor-product or box·spline) mesh. A mesh is vertex-degree bounded if at most four cells meet at each vertex, and polyhedral if all cells with more than four edges are planar.
t Department of Computer Science, Purdue University, W-Lafayette IN 47907-1398 Supported by NSF grant CCR·9211322
1
J
P~l~ing polyhed.
S~P'
1593
L Introduction
Recent advances in the theory of splines over irregular meshes suggest that the degree and number of polynomial pieces necessary to smooth a mesh of points are related both
to the combinatorial structure and the geometric variation of the mesh. lvIesh cells are considered to have low variation if they are either planar or quadrilateral, and the combinatorial structure of a mesh is simpler where the vertex degree, the number of neighbors to the mesh point, is uniform or four. Thus, we have the following classification. The mesh type of least complexity is the
well~known
regular mesh. In a regular mesh each
mesh point is surrounded by four I possibly triangulated, quadrilateral cells. Such a mesh can be viewed as the control mesh of a tensor product spline or a four direction box spline yielding one biquadratic piece in the case of a C 1 tensor-product construction and four quadratic pieces per cell in the case of a C 1 box spline surface (see e.g. [1]). At the other end of the spectrum are irregular meshes. Irregular meshes may have any number of not necessarily planar cells surrounding mesh points and an arbitrary number of vertices per cell. To meet the primary goal, low polynomial degree, various approaches adopt a refinement strategy to partially regularize the irregular mesh and triangulate it to reduce the geometric variation. For example, [13] and [14] triangulate and split the mesh to fit piecewise quartic surfaces and (8] restricts the combinatorial structure of the input mesh to fit S-patches. The construction in [9] (see also [10] and [7] for a similar construction based on three-sided pieces) refines the mesh into a simpler mesh to obtain on the average 16 biquadratic or bicubic polynomial pieces per original cell. In [11] it was observed that degree-bounded meshes are almost as general as irregular meshes but can be smoothed using only one refinement and hence 4 pieces per facet. An irregular mesh is degree-bounded if cells with more than four edges have no vertices with more than four neighbors and, symmetrically, vertices with more than four neighbors are surrounded by cells with at most four edges. This paper completes the 'complexity scale of meshes' by showing that vertex-degree bounded polyhedral meshes can be smoothed without further refinement. The proof is an 2
J
P~'~r&
Srnoo'hinll:
polyh~dr..
algorithm that constructs the surface using four total degree cubic patches per facet. (An analogous tensor product scheme would be akin to [5] and (12].) Vertex-degree bounded
polybedral mesbes have at most four cells meet at each vertex and cells with more than four edges are planar. The planarity requirement for cells with more than four edges is a major restriction since it prevents us in general from simply using the dual of a triangulation. Yet, there are important classes of polyhedral meshes. First the DooSabin [3] or Catmull-Clark [2] refinement used to smooth irregular meshes turns them into special vertex-degree bounded meshes, that become polyhedral after local affine projection. Secondly, meshes may be modified to have the required structurej for example by chopping off convex vertices. Finally, buckminsterfullerene geodesic structures found in nature [15} I like the dodecahedron below obey the constraint (and partially motivated this study).
,, )..
,, , >-
,
The planarity of the cells allows both a simple construction and characterization of the surface. The algorithm in Section 3 uses the planar mesh cells as tangent planes and allows the user to choose an interpolation point per cell. Should the mesh additionally be regular, then the Cl surface becomes the quadratic defined by the tangent planes.
2. A hierarchy of meshes The relationships between the mesh types are as follows.
An irregular mesh can
be transformed into a degree-bounded mesh by triangulating offending cells. A degreebounded mesh transforms to a vertex-regular polyhedral mesh after one Doo-Sabin refinement step [3] followed by a projection. Similarly, two subdivisions may be used to refine
3
Smoo'hing
p"lyh~d
Purdue e-Pubs Computer Science Technical Reports
Department of Computer Science
1993
Smoothing Vertex-degree Bounded Polyhedra Jörg Peters Report Number: 93-051
Peters, Jörg, "Smoothing Vertex-degree Bounded Polyhedra" (1993). Computer Science Technical Reports. Paper 1065. http://docs.lib.purdue.edu/cstech/1065
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information.
SMOOTHING VERTEX·DEGREE BOUNDED POLYHEDRA
Jorg Peters
CSD-TR-93-6S1 August 1993
Smoothing vertex-degree bounded polyhedra
by Jorg Peters t [email protected]
Key words: C 1 surface, corner cutting, box splines, C 1 spline mesh, blending Running title: Smoothing polyhedra Version: Sept 15 93
Date printed: October 13, 1993
Submitted to: ACM TOG
Abstract A special class of meshes for outlining a surface, vertex-degree bounded, polyhedral meshes, is shown to have a simple smoothing algorithm that generates a small number of low degree polynomial pieces per cell. In particular, the input mesh need not be refined to obtain the Bernstein-Bezier control points as averages of the mesh points. Thus vertexdegree bounded, polyhedral meshes are the simplest and immediate generalization of the regular (tensor-product or box·spline) mesh. A mesh is vertex-degree bounded if at most four cells meet at each vertex, and polyhedral if all cells with more than four edges are planar.
t Department of Computer Science, Purdue University, W-Lafayette IN 47907-1398 Supported by NSF grant CCR·9211322
1
J
P~l~ing polyhed.
S~P'
1593
L Introduction
Recent advances in the theory of splines over irregular meshes suggest that the degree and number of polynomial pieces necessary to smooth a mesh of points are related both
to the combinatorial structure and the geometric variation of the mesh. lvIesh cells are considered to have low variation if they are either planar or quadrilateral, and the combinatorial structure of a mesh is simpler where the vertex degree, the number of neighbors to the mesh point, is uniform or four. Thus, we have the following classification. The mesh type of least complexity is the
well~known
regular mesh. In a regular mesh each
mesh point is surrounded by four I possibly triangulated, quadrilateral cells. Such a mesh can be viewed as the control mesh of a tensor product spline or a four direction box spline yielding one biquadratic piece in the case of a C 1 tensor-product construction and four quadratic pieces per cell in the case of a C 1 box spline surface (see e.g. [1]). At the other end of the spectrum are irregular meshes. Irregular meshes may have any number of not necessarily planar cells surrounding mesh points and an arbitrary number of vertices per cell. To meet the primary goal, low polynomial degree, various approaches adopt a refinement strategy to partially regularize the irregular mesh and triangulate it to reduce the geometric variation. For example, [13] and [14] triangulate and split the mesh to fit piecewise quartic surfaces and (8] restricts the combinatorial structure of the input mesh to fit S-patches. The construction in [9] (see also [10] and [7] for a similar construction based on three-sided pieces) refines the mesh into a simpler mesh to obtain on the average 16 biquadratic or bicubic polynomial pieces per original cell. In [11] it was observed that degree-bounded meshes are almost as general as irregular meshes but can be smoothed using only one refinement and hence 4 pieces per facet. An irregular mesh is degree-bounded if cells with more than four edges have no vertices with more than four neighbors and, symmetrically, vertices with more than four neighbors are surrounded by cells with at most four edges. This paper completes the 'complexity scale of meshes' by showing that vertex-degree bounded polyhedral meshes can be smoothed without further refinement. The proof is an 2
J
P~'~r&
Srnoo'hinll:
polyh~dr..
algorithm that constructs the surface using four total degree cubic patches per facet. (An analogous tensor product scheme would be akin to [5] and (12].) Vertex-degree bounded
polybedral mesbes have at most four cells meet at each vertex and cells with more than four edges are planar. The planarity requirement for cells with more than four edges is a major restriction since it prevents us in general from simply using the dual of a triangulation. Yet, there are important classes of polyhedral meshes. First the DooSabin [3] or Catmull-Clark [2] refinement used to smooth irregular meshes turns them into special vertex-degree bounded meshes, that become polyhedral after local affine projection. Secondly, meshes may be modified to have the required structurej for example by chopping off convex vertices. Finally, buckminsterfullerene geodesic structures found in nature [15} I like the dodecahedron below obey the constraint (and partially motivated this study).
,, )..
,, , >-
,
The planarity of the cells allows both a simple construction and characterization of the surface. The algorithm in Section 3 uses the planar mesh cells as tangent planes and allows the user to choose an interpolation point per cell. Should the mesh additionally be regular, then the Cl surface becomes the quadratic defined by the tangent planes.
2. A hierarchy of meshes The relationships between the mesh types are as follows.
An irregular mesh can
be transformed into a degree-bounded mesh by triangulating offending cells. A degreebounded mesh transforms to a vertex-regular polyhedral mesh after one Doo-Sabin refinement step [3] followed by a projection. Similarly, two subdivisions may be used to refine
3
Smoo'hing
p"lyh~d
